I don't know specifically about Texas and Louisiana, but I've sued several ties in small claims court (and won).

Do an online search for "small claims court" for your county or parish. You very likely will find the information you're looking for: What the maximum amount you can sue for is (it varies), whether you need a lawyer (usually not, though corporations often are required to have legal representation), the paperwork involved, the steps you have to go through etc.

A few tips: Make sure you follow the specified procedures exactly. And have good, complete documentation of what occurred and the compensation you're seeking. And make sure that if the court first requires other actions (such as notification to the moving company of your grievances), follow those procedures to the letter. And carefully read your contract with the moving company to make sure you haven't signed away rights, and to make sure that they have violated their agreement with you.

Many people have sued moving companies and won. If your claim is legitimate, you'll have a good chance, too.